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Deriving Macroscopic Myocardial Conductivities by Homogenization of Microscopic Models

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Abstract

We derive the values for the intracellular and extracellular conductivities needed for bidomain simulations of cardiac electrophysiology using homogenization of partial differential equations. In our model, cardiac myocytes are rectangular prisms and gap junctions appear in a distributed manner as flux boundary conditions for Laplace’s equation. Using directly measurable microproperties such as cellular dimensions and end-to-end and side-to-side gap junction coupling strengths, we inexpensively obtain effective conductivities close to those given by simulations with a detailed cyto-architecture (Stinstra et al. in Ann. Biomed. Eng. 33:1743–1751, 2005). This model provides a convenient framework for studying the effect on conductivities of aligned vs. brick-like arrangements of cells and the effect of different distributions of gap junctions along the myocyte membranes.

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References

  • Camelliti, P., Green, C., Kohl, P., 2006. Structural and functional coupling of cardiac myocytes and fibroblasts. In: Dhein, S. (Ed.), Cardiovascular Gap Junctions, Adv. Cardiol., vol. 42, pp. 132–149. Karger, Basel.

    Chapter  Google Scholar 

  • Clerc, L., 1976. Directional differences of impulse spread in trabecular muscle from mammalian heart. J. Physiol. 255, 335–346.

    Google Scholar 

  • Fishman, G., 2008. Personal communication.

  • Henriquez, C., 1993. Simulating the electrical behavior of cardiac tissue using the bidomain model. Crit. Rev. Biomed. Eng. 21, 1–77.

    Google Scholar 

  • Hubbard, M., Ying, W., Henriquez, C., 2007. Effect of gap junction distribution on impulse propagation in a monolayer of myocytes: a model study. Europace 9, vi20–vi28.

    Article  Google Scholar 

  • Keener, J., 1996. Direct activation and defibrillation of cardiac tissue. J. Theor. Biol. 178, 313–324.

    Article  Google Scholar 

  • Keener, J., Sneyd, J., 1998. Mathematical Physiology. Springer, Berlin.

    MATH  Google Scholar 

  • Kléber, A., Riegger, C., 1987. Electrical constants of arterially perfused rabbit papillary muscle. J. Physiol. 385, 307–324.

    Google Scholar 

  • Kohl, P., Camelliti, P., Vurton, F., Smith, G., 2005. Electrical coupling of fibroblasts and myocytes: relevance for cardiac propagation. J. Electrocardiol. 38, 45–50.

    Article  Google Scholar 

  • Krassowska, W., Pilkington, T., Ideker, R., 1990. Potential distribution in three-dimensional periodic myocardium—Part I: solution with two-scale asymptotic analysis. IEEE Trans. Biomed. Eng. 37(3), 252–266.

    Article  Google Scholar 

  • Mori, Y., Fishman, G., Peskin, C., 2008. Ephaptic conduction in a cardiac strand model with 3D eletrodiffusion. Proc. Natl. Acad. Sci. 105, 6463–6468.

    Article  Google Scholar 

  • Neu, J., Krassowska, W., 1993. Homogenization of syncytial tissues. Crit. Rev. Biomed. Eng. 21, 137–199.

    Google Scholar 

  • Panfilov, A., Keldermann, R., Nash, M., 2007. Drift and breakup of spiral waves in reaction–diffusion-mechanics systems. Proc. Natl. Acad. Sci. 104, 7922–7926.

    Article  Google Scholar 

  • Roberts, D., Scher, A., 1982. Effect of tissue anisotropy on extracellular potential fields in canine myocardium in situ. Circ. Res. 50, 342–351.

    Google Scholar 

  • Roberts, D., Hersh, L., Scher, A., 1979. Influence of cardiac fiber orientation on wavefront voltage, conduction velocity, and tissue resistivity in the dog. Circ. Res. 44, 701–712.

    Google Scholar 

  • Shaw, R., Rudy, Y., 1997. Ionic mechanisms of propagation in cardiac tissue—roles of the sodium and L-type calcium currents during reduced excitability and decreased gap junction coupling. Circ. Res. 81, 727–741.

    Google Scholar 

  • Stinstra, J., Hopenfeld, B., Macleod, R., 2005. On the passive cardiac conductivity. Ann. Biomed. Eng. 33, 1743–1751.

    Article  Google Scholar 

  • Weidmann, S., 1970. Electrical constants of trabecular muscle from mammalian heart. J. Physiol. 210, 1041–1054.

    Google Scholar 

  • Yao, J., Gutstein, D., Liu, F., Fishman, G., Wit, A., 2003. Cell coupling between ventricular myocyte pairs from connexin43-deficient murine hearts. Circ. Res. 93, 736–743.

    Article  Google Scholar 

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Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, New York, NY, USA

    Paul E. Hand, Boyce E. Griffith & Charles S. Peskin

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  1. Paul E. Hand
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  2. Boyce E. Griffith
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  3. Charles S. Peskin
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Correspondence to Paul E. Hand.

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Hand, P.E., Griffith, B.E. & Peskin, C.S. Deriving Macroscopic Myocardial Conductivities by Homogenization of Microscopic Models. Bull. Math. Biol. 71, 1707–1726 (2009). https://doi.org/10.1007/s11538-009-9421-y

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  • DOI: https://doi.org/10.1007/s11538-009-9421-y

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